Spacetimes with changing signature in General Relativity
Tzitzimpasis Paraskevas
A thesis submitted in partial fulfillment for the bachelor in Physics
in the
Department of Physics Aristotle University of Thessaloniki
Supervisor: Prof. Christos Tsagas
July 2016
Contents
1 Introduction5
2 Fundamentals7 2.1 The 1+3 formalism ...... 7 2.2 The FRW universe...... 8 2.3 The transition formulation ...... 10 2.3.1 General formulation...... 10 2.3.2 Notation in FRW metrics...... 12
3 Evolution of the Universe 15 3.1 Solution of the Friedmann equations for a fluid with constant barotropic index . 15 3.1.1 Lorentzian Regime...... 15 3.1.2 Euclidean Regime...... 17 3.2 General properties of the solutions...... 18 3.2.1 Existence of an extremum ...... 18 3.2.2 A simple form of the second junction condition...... 18 3.3 Application for a relativistic fluid...... 20 3.3.1 Lorentzian regime...... 20 3.3.2 Euclidean regime...... 21 3.3.3 Matching the solutions for k=+1 on both sides...... 21 3.3.4 Matching the solutions for k=-1 on both sides...... 23 3.4 Application for inflationary fluid...... 24 3.4.1 Lorentzian regime...... 25 3.4.2 Euclidean regime...... 25 3.4.3 Matching the solutions for k=+1...... 26 3.4.4 Matching the solutions for k=-1 ...... 26
4 Conclusion 29
3 A The matter tensor 31
B The field equations 33
C The Friedmann equations 35
4 Chapter 1
Introduction
Our standard cosmological models describe gravitation in terms of the General Theory of Relativity, formulated in the last century. The theory of Relativity has had a remarkable impact in our understanding of the universe. Its confirmations begin with Einstein’s paper for the precession of Mercury’s perihelium and come all the way to the recent detection of gravitational waves. Although we should feel grateful for having such an effective and elegant tool in our hands, there are some good reasons to be yet unsatisfied. From our current understanding of the physical world, we expect quantum mechanics to have a significant interference at very large energies or very small spacetime scales. Common examples of physical phenomena where quantum mechanics is expected to dominate over our classical picture are black holes singularities or the very beginning of our universe. Our cosmological models can take into account the quantum behaviour of the matter content of the universe but they fail when it comes to the interplay of the quantum theory and gravity. This being said, it should be no surprise that those models become ill-defined at the big bang singularity where they predict infinities in quantities like the curvature or the density. A theory of quantum gravity would be necessary to describe early times preceding inflation. It would also be necessary to eliminate the singularity of the big bang. There have been many attempts to formulate a quantum theory of gravitation sometimes by modifying Classical Relativity and sometimes by building it from zero. One basic difficulty occurs because quantum theory, including modern gauge field theories, usually treats fields on a fixed background, either Euclidean space, in non-relativistic quantum mechanics, or Minkowski spacetime, in relativistic quantum theory. The formulation is thus not well adapted to considering the situation where the background metric is itself a field variable. Even if we had a complete theory of quantum gravity, we would still need to know the initial condition of our universe in order to understand its evolution. A famous proposal due to James Hartle and Stephen Hawking is the so called ”Hartle-Hawking proposal”. One very fascinating aspect of this proposal is the idea that the signature of the metric tensor should change at the very early stages of the universe when quantum effects become so dominant that they affect the very nature of space and time. In particular, it is assumed that spacetime is no longer Lorentzian
5 which means havinga metric signature (-+++) but ”Euclidean” (or sometimes called Riemannian) in the sense that its purely spatial and the signature of the metric tensor is (++++). The use of a Euclidean ”spacetime” allows us to assume that the (purely spatial) universe in its early state had the topology of a 4-sphere. By making this assumption, there is an elegant way to avoid the need of a beginning of our universe in the sense that the universe is compact. A simple way to picture this argument is if we imagine a 2-dimensional sphere, say the Earth. If we start moving in a certain direction, for example south, we will reach a point , the south pole, where there in no meaning in asking ”what lies south of the south pole”. In the same sense, if we turn time into a spatial direction, we can create a universe in which the question ”what was the universe like before that time” has no meaning. The idea of Hartle and Hawking is not necessarily correct but it surely provides us with new possibilities that have both scientific and philosophical significance. This idea of a potential signature change has been adopted by others and applied in Classical Relativity. Now one might be suspicious about a solution of the Einstein field equations with a signature change. Besides, it is very common in General Relativity to assume that the metric signature is constant and then solve the field equations. However, this is more of an assumption we impose, rather than a real physical demand. When we try to apply the idea of signature change in a cosmological model we are faced with a significant problem. Namely the metric at the hypersurface of transition has a zero eigenvalue and thus becomes degenerate. This is clearly a problem considering that in General Relativity the metric is assumed to be non-degenerate. However, this can be tackled by adopting a careful treatment of the transition. The metric is well defined in the rest of the manifold and so are the field equations. What we will attempt to do is solve the field equations in the Lorentzian and Euclidean regime and impose the appropriate junction conditions on the surface of change. This will produce new solutions of the field equations with new degrees of freedom and consequently new, interesting properties.
6 Chapter 2
Fundamentals
2.1 The 1+3 formalism
Let O be an open region in spacetime (M, g). A congruence in O is a family of curves such that every P O belongs to one and only one curve from this family. We will be discussing timelike congruences i.e. families of timelike geodesics that fill spacetime and do not intersect. We can imagine such a congruence as a fluid, every point of which corresponds to an observer moving along the curves xa = xa(τ) where τ is the proper time of the observer. Obviously the generator of the congruence will be the tangent vector field dxa ua = (2.1) dτ a which is the 4-velocity of the observer normalized so that u ua = −1. The existence of a preferred direction in spacetime at each point (defined by the 4-velocity mentioned above), implies the existence of simultaneous rest hypersurfaces of dimensionality 3. These hypersurfaces have the structure of a manifold embedded in M and thus there is a natural way to define a metric on them. There is a natural map from the embedded hypersurface Σ to the manifold M by mapping every point of Σ to itself in M. This map can be used to construct the pullback of the metric g to Σ.It is easily verified that the metric
hab = gab + uaub (2.2) satisfies the following relations:
a b a a a b h bh c = h c, h a = 3, h bu = 0 (2.3)
a which indicate that hab is the object we are looking for and that additionally h b is the projection tensor projecting into the three-dimensional tangent plane orthogonal to ua. It is well known in the study of fluid dynamics that the evolution of a fluid is characterized by the tensor uab = ua;b which determines the evolution of a deviation vector of the deformable medium.More specifically, ua;b can be decomposed in the following convenient way: 1 u = σ + ω + Θh − A u (2.4) a;b ab ab 3 ab a b
7 c d where the three first terms are the "spatial derivative" ha hb uc;d while the last term corresponds to the "time derivative" u˙ aub. From the above quanities, Θ is the fractional change of volume per unit time.When Θ > 0, the fluid expands while when Θ < 0, the fluid contracts.We shall use Θ to define the scale factor from the equation a˙ 1 = Θ (2.5) a 3
The rotation tensor ωab changes the orientation of the deformable medium and being an- tisymmetric, we can define a vector with its independent components in the standard way:
1 ω = ωbc (2.6) a 2 abc
The shear tensor σab carries the information about the change in shape of the fluid. It is a symmetric tensor with zero trace. The last term is the 4-acceleration of the observer which vanishes in the case of a congruence a of geodesics in light of the geodesic equation u ub;a = 0. Intuitively this is just the statement that non-accelerating observers move on geodesics.
2.2 The FRW universe
In this section we will present the FRW model which is the one that will be used in this paper. This model contains two fundamental (and very strong) assumptions: Homogeneity and Isotropy. By isotropy we mean that there is no preferred direction in space.If we want to express this intuitive concept of isotropy in a formal way, we can say that a manifold M is isotropic around a point P M if for any two vectors V, W TP (M) there is an isometry of M such that the pushforward of W is parallel with V. By homogeneity we mean that the space has the same properties in every point.In a more formal language, a metric space is homogenous if there exist isometries that carry any point XM to any other point in M. A space is said to be maximally symmetric if one the (equivalent) following conditions holds:
• The space is isotropic about some point and homogenous.
• The space is isotropic about any point.
These spaces are called maximally symmetric because they posses the maximum number of linearly independent Killing vectors which is N(N+1)/2 (which is the general definition of a maximally symmetric space). This maximal symmetry has many interesting implications for the structure of the metric space. First of all the Riemann tensor can only have the following form:
R R = g g − g g (2.7) abcd N(N − 1) db ac cb ad
8 which by contraction with the metric tensor gac gives the Ricci tensor: R R = g (2.8) db N db and using the Bianchi identity: 1 Rab − gabR = 0 (2.9) 2 ;a and the fact that the metric tensor is covariantly constant we arrive at: 1 1 ∂ − R = 0 (2.10) N 2 ∂xa The above result is of great importance. It indicates that in a space where (2) holds everywhere (as in the case of the maximally symmetric spaces that we are investigating) the Ricci scalar is constant. An additional factor that makes the study of maximally symmetric spaces easier is the following powerful theorem.
Theorem 1 Given two maximally symmetric metrics with the same scalar curvature K and the same numbers of eigenvalues of each sign, it will always be possible to find a coordinate transformation that carries one metric into the other.
Now that we have an idea of the extent to which the assumptions of homogeneity and isotropy simplify the study of a metric space, we may proceed with more details about the FRW metrics. By observations we know that what we call "space" is homogenous and isotropic at large enough scales.Among others, a very strong reason to believe this, is the data we have about the Cosmic Microwave Background which exhibits a remarkable isotropy.In order to be able to use this information we have about the three dimensional space we live in, we use the 1+3 foliation of spacetime which allows us to handle space and time separately. The three dimensional,spacelike sub-manifolds Σ that foliate spacetime are thus maximally symmetric but spacetime as a whole is not. Thus, the complete metric takes the form:
ds2 = −dt2 + R2(t)dσ2 (2.11) where t is an arbitrary coordinate of the timelike curves, R(t) is the scale factor and dσ2 is the metric on the hypersurfaces Σ. The coordinate system that we use here is the one where the symmetries are most profoundly manifested. In this particular system, the cross terms dtdui disappear and gtt = −1.These coordinates are called comoving (or Lagrangian) and they are widely used in cosmology.They are constructed in the following way:
• We choose one of the three dimensional hypersurfaces Σ0 and label each timelike curve of our congruence with the three numbers that correspond to the point of intersection
(u1, u2, u3).
9 • We then extend this labelling off the hypersurface Σ0 by maintaining the same labelling off the hypersurface.
• We finally define a time coordinate along the timelike curves (for example proper time).
We now emphasize on the positive definite and maximally symmetric metric of Σ which can be expressed in the following form:
2 a b dσ = habdu du (2.12)
Using the fact that maximal symmetry also implies spherical symmetry which corresponds to the Schwarzschild metric and taking into account the form of the Ricci tensor given by (2) we conclude that it is possible to choose coordinates r, θ, φ, t for which the metric takes the form:
dr2 dσ2 = + r2dθ2 + r2 sin2 θdφ2 (2.13) 1 − kr2 and the complete metric (5) takes the form:
dr2 ds2 = −dt2 + R2(t) + r2dθ2 + r2 sin2 θdφ2 (2.14) 1 − kr2
where k is the normalized curvature that takes the discrete values {−1, 0, 1}.
When k=-1 Σ has a negative curvature and is called open. When k=0 the hypersurface is flat and has the geometry of Euclidean space and finally when k=1 the space has positive curvature and is closed.Examples of spaces with such curvatures are the hyperboloid(k=-1),the ordinary three dimensional flat space (k=0) and the sphere (k=1).The study of only those three spaces imposes no loss of generality because Theorem 1 ensures us that every other space will be isomorphic to one of the above three.
2.3 The transition formulation
2.3.1 General formulation
Let us consider a manifold M which is divided in two regions M + and M − with different metrics ± gab and different signatures (− + ++) and (+ + ++) . Those two regions are separated by a hypersurface Σ which is assumed to have a well defined, non-degenerate induced metric hab (this implies that Σ cannot be null). In practice we will only use timelike or spacelike hypersurfaces. A natural question that arises at such cases is how we can join those two metrics so that the resulting metric satisfies the Einstein equations. First we choose a coordinate system xa in a region around Σ. We use the coordinate l to measure the ”distance” from Σ. So l can be proper time or proper distance of a chosen congruence of timelike or spacelike geodesics that pass through Σ. We define l = 0 on Σ and we
10 also take l to be positive in M + and negative in M − Since Σ is given by the equation l = 0, the normal vector will be given by the gradient of l:
na = −∂al (2.15)
where = 1 corresponds to the Lorentzian regime with signature (− + ++) and = 1 to the Euclidean with signature (++++). We also introduce the standard notation for the discontinuity of a quantity F that is defined on both sides of Σ:
[F ] = lim F − lim F (2.16) l→0+ l→0−
If we define the step function Θ(l) which satisfies Θ(l) = 0 for l < 0, Θ(l) = 1 for l > 0 and undefined at l = 0 then the complete metric can be written as
+ − gab = Θ(l)gab + Θ(−l)gab (2.17)
Differentiation of (2.17) gives:
+ gab,c = Θ(l)gab,c + Θ(−l)gab,c + δ(l)[gab]nc (2.18)
which implies that in order to have well defined derivatives of the metric (and consequently well defined connection) we must impose continuity of the metric (in this specific coordinate system)
[gab] = 0 (2.19)
However in the presence of signature change, continuity of the metric is not compatible with ± its non-degeneracy. Thus, the metric becomes degenerate on Σ(detg |Σ= 0). Nevertheless the ”singular” eigenvector is the one in the direction orthogonal to Σ so the induced metric is still well defined and satisfies the first junction condition (which can be easily shown to be coordinate idependent)
[hab] = 0 (2.20)
The second junction condition is a little harder to attain. In the same way as above, we calculate the Riemann tensor
a +a −a a R bcd = Θ(l)R bcd + Θ(−l)R bcd + δ(l)A bcd (2.21)
where a a a A bcd = [Γ bd]nc − [Γ bc]nd (2.22)
And by calculating the Einstein tensor and using the field equations, the form of the stress tensor turns out to be + − Tab = Θ(l)Tab + Θ(−l)Tab + δ(l)Sab (2.23)
11 where 1 8πS ≡ A − Ag (2.24) ab ab 2 ab a a Aab = A bad,A = A a (2.25)
Now one might think that the reasonable thing to do is demand that the singular term in the stress tensor vanishes. However, we can treat Sab as a surface stress tensor describing the fluid in a thin shell of infinitesimal thickness. Thus the above treatment only makes sense if Σ is actually a very thin shell rather than an ordinary hypersurface. The three-tensor Sab can also be expressed in terms of the jump in extrinsic curvature giving what is known as Lanczos equation
S = − [K ] − [K]h (2.26) ab 8π ab ab
Of course we see that if the surface had no thickness, that would immediately imply that the jump in extrinsic curvature would be zero and Sab would disappear from our equations as expected. The condition
[Kab] = 0 (2.27) is commonly used as the second junction condition that completes the set of equations we need to connect the two metrics g±. In our study, we will not restrict ourselves to considering transition hypersurfaces of constant extrinsic curvature.
2.3.2 Notation in FRW metrics
We are seeking for an easy formalism that will allow us to alternate between the standard lorentzian spacetime with signature (− + ++) and the positive-definite Euclidean with signature (++++).At the changing surface, the timelike congruence will become spacelike so the norm of the generator ua will be altered from -1 to +1. This fact can be easily expressed in the following way:
a 2 uau = −, = 1 (2.28) where is a scalar which is constant in the regions M + and M − but not in the manifold M as a a whole. The induced metric hab is also going to change considering that it depends on u . More specifically we have:
hab = gab + uaub (2.29)
b which is easily verified by taking the inner product of hab with u which should identically vanish. The FRW metric will also change by the same factor of and will become:
dr2 ds2 = −dt2 + R2(t) + r2dθ2 + r2 sin2 θdφ2 (2.30) 1 − kr2
12 In FRW geometries, the rotation tensor ωa vanishes which implies (by Frobenius theorem) that ua is the gradient of a scalar. In our case, the surfaces orthogonal to ua are those of constant t which means that: a a ua = −t,a, u = δ0 (2.31)
13
Chapter 3
Evolution of the Universe
In the present chapter we will attempt to solve the set of equations (A.4), (C.5) and (C.9) namely
µ˙ + (µ + p)Θ = 0 (conservation equation) (3.1)
R¨ κ 3 = µ + 3p (Raychaudhuri equation) (3.2) R 2
R˙ 2 κ κ = µ − (F riedmann equation) (3.3) R2 3 R2 As pointed out in Appendix C, these equations are not independent. What is more, in order to solve them and find the scale factor as a function of the time coordinate t, we need to specify an equation of state for the fluid we are considering. Since we are dealing with barotropic fluids, the equation of state will have the form
p = p(µ) (3.4)
3.1 Solution of the Friedmann equations for a fluid with constant barotropic index
We will restrict ourselves to the case of a constant barotropic index, namely we consider equations of state of the form p = wµ (3.5)
.
3.1.1 Lorentzian Regime
By using equations (3.2),(3.3),(3.5) we can easily eliminate µ and p from the equations (which is quite obvious since our set of equations is linear with respect to those variables) and end up with
15 a differential equation for the scale factor.Doing this we get
RR¨ + MR˙ 2 + Mk = 0 (3.6) where we have defined 1 M = 1 + 3w (3.7) 2 since the variable t is missing from the equation, we can easily reduce its order by setting dρ R˙ = ρ, R¨ = ρ dR and the above equation simplifies to dρ ρR + M(ρ2 + k) = 0 dR which can be directly integrated to give c R˙ 2 + k = 1 , c > 0 (3.8) R2M 1 The above equation can be integrated as well and give us an expression for the function t = t(R):
• For k=0 integration gives (for M 6= −1) M+1 q R 1 M+1 √ √ = t + c2 ⇒ R = (M + 1) c1(t + c2) (3.9) M + 1 c1
• For k=-1 (3.8) gives
1 1 1 c R F , − ; 1 − ; − 1 = t + c (3.10) 2 1 2 2M 2M R2M 2
where 2F1 is the Gauss Hypergeometric Function.
• For k=+1 (3.8) gives
1 1 1 c ıR˙ F , − ; 1 − ; 1 = t + c (3.11) 2 1 2 2M 2M R2M 2
Now we should carefully look at this result. First of all, in order to carry out the integration we notice that (3.8) for k=+1 implies that
c1 √ > 1 ⇒ R < 2M c (3.12) R2M 1 sinceR˙ cannot be imaginary. Something that causes some discomfort by looking at (3.11)
is the fact that ordinary Gauss Hypergeometric Function 2F1 is defined when the norm of its argument is less than unit, while it is clear from (3.12) that this condition is not
satisfied in our case. This is why 2F1 here denotes the analytically continued Gauss Hypergeometric Function which can be defined for all real numbers (we are not interested
16 in complex arguments here). This also holds for (3.10) where the argument takes arbitrary (negative) values. Something else that could be unexpected is the presence of the imaginary unit in (3.11) where we would normally expect only real numbers to appear. By careful examination of (3.11) we conclude that the right hand side is indeed a complex function which nevertheless has a constant imaginary part that vanishes with the proper selection of
the integration constant c2 which is complex as well. So what we will be left with is a real function (as it should be). By using the properties of analytically continued hypergeometric functions, we can show that (3.11) is equivalent to Γ 1 − 1 Γ − 1 − 1 1 1 1 3 1 R2M RM+1 2M 2M 2 F , + ; + ; √ = t + c (3.13) 1 1 1 2 1 2 2 2M 2 2M c c 2 Γ − 2M Γ 2 − 2M 1 1 which is apparently real and what is more, the hypergeometric function appearing here is the usual one which is well defined as can be seen by looking at (3.12).
3.1.2 Euclidean Regime
The case for the Euclidean regime is very similar to the above discussion. We consider an equation of state like (3.5) but we do not in general assume that the barotropic index is constant along the transition hypersurface. We denote the index in the positive definite spacetime with z and thus the equation of state is p = zµ (3.14) Just like previously, we can find a differential equation for the scale factor. This turns out to be
RR¨ + NR˙ 2 − Nk = 0 (3.15) where 1 N = 1 − 3z (3.16) 2 The similarities with (3.6) are profound. The only differences are that the constant M is replaced with N and there is an extra minus in (3.15). A significant conclusion that can be obtained by comparing (3.6) with (3.15) is that if the equation of state is of the form p = wµ (in which case N = M), then the solutions for k=+1 in the Euclidean spacetime are the same as solutions for k=-1 in the Lorentzian regime and vice versa. The solutions for k=0 are the same in both spacetimes. Using the same method to reduce the order of (3.15) we get c0 R˙ 2 − k = 1 , c0 > 0 (3.17) R2N 1 and the solutions will have the same form as before (but interchanged for k=±1)
• For k=0 (3.17) gives N+1 q R 1 N+1 p = t + c0 ⇒ R = (N + 1) c0 (t + c0 ) (3.18) p 0 2 1 2 N + 1 c1
17 • For k=-1 we get 1 1 1 c0 Rı˙ F , − ; 1 − ; 1 = t + c0 (3.19) 2 1 2 2N 2N R2N 2 which becomes as before
Γ 1 − 1 Γ − 1 − 1 1 1 1 3 1 R2N RN+1 2N 2N 2 F , + ; + ; = t + c0 (3.20) 1 1 1 2 1 2 2 2N 2 2N c p 0 2 Γ − 2N Γ 2 − 2N 1 c1
• For k=+1 we get
1 1 1 c0 R F , − ; 1 − ; − 1 = t + c0 (3.21) 2 1 2 2N 2N R2N 2
3.2 General properties of the solutions
3.2.1 Existence of an extremum
There are some properties of the above solutions that are worth mentioning. First of all, let us assume that the curvature is constant in spacetime. If it is zero, then the solutions (3.9) and (3.18) govern the evolution of the scale factor and we see that the whole spacetime is flat. However if we start with a Euclidean spacetime of curvature ±1 (which will be the same in the Lorentzian regime) the pairs of equations (3.10) and (3.20) or (3.11) and (3.21) indicate that there is a change in some fundamental properties. If we start with an open Euclidean spacetime we could end in a closed Lorentzian after the transition and vice versa. This is because from equations (3.8) and (3.17) we can see that the existence of a possible extremum depends on the product k.
3.2.2 A simple form of the second junction condition
Another very important property arises from the second junction condition. In practice, in order to attain a valid form of the second junction condition we use the Friedmann equation
R˙ 2 κ k = µ − (3.22) R2 3 R2 By taking the jump of the two sides of this equation we get:
R˙ 2 κ k [ ] = [µ] − [ ] R2 3 R2 where the jump of is obviously [] = 1 − (−1) = 2
Now assuming that the density µ is the same on both sides of the transition hypersurface Σ we can impose [µ] = 0
18 and thus we arrive at R˙ 2 k [ ] = −[ ] R2 R2 where by using the standard properties of limits and the first junction condition (continuity of scale factor) it is easy to see that [R˙ 2] = −[k] (3.23) and if k is constant (which is reasonable to assume in most cases)
[R˙ 2] = −2k so we have two possibilities
• If k=0 then [R˙ 2] = 0 (3.24) from which it is obvious that the derivative of the scale factor is also continuous.
• If k=+1 then [R˙ 2] = −2 (3.25) ˙ ˙ so in general R |E> R |L.
• If k=-1 then [R˙ 2] = 2 (3.26) ˙ ˙ so R |E< R |L.
Equations (3.24),(3.25) and (3.26) together form the second junction conditions for the cases of constant k. So an obvious way is to find the form of the function R˙ and impose those conditions on it. However there is a much easier way to proceed. By subtracting equation (3.17) from (3.8) (or to be more accurate their limits approaching Σ) we can easily see that 0 ˙ 2 c1 c1 [R ] + [k] = 2M − 2N R0 R0 and using (3.23) we get 0 c1 c1 2M = 2N (3.27) R0 R0 where R0 is the limiting value of the scale factor on both sides (which have to be equal according to the first junction condition). Equation (3.27) is a very convenient and elegant expression of the second junction condition. It is important to note that in the case of an equation of state of the form p = wµ where w is the barotropic index (constant through the whole spacetime), equation (3.27) reduces to the profoundly simple statement that
0 c1 = c1 (3.28) because in that case M = N as can be seen from equations (3.7) and (3.16).
19 3.3 Application for a relativistic fluid
We here consider the case where the equation of state is given by 1 p = µ 3 which preserves the zero trace of the stress tensor of the barotropic fluid. The general set of equations is:
3R˙ µ˙ + µ + p = 0 (3.29) R 3R¨ κ = − µ + 3p (3.30) R 2 R˙ 2 κ k = µ − (3.31) R2 3 R2 1 p = µ (3.32) 3 where = 1 in Lorentzian spacetime and −1 in Euclidean.
3.3.1 Lorentzian regime
In the Lorentzian regime if we solve the set of equations for a general barotropic index w, we arrive at the differential equation c R˙ 2 + k = 1 (3.33) R2M where 1 M = 1 + 3w 2 so for w = 1/3 as in the case of a relativistic fluid, M = 1 and thus this can be integrated to give
• For k=0 (flat) R2 √ = c t + c 2 1 2
q √ R = 2( c1t + c2) (3.34)
• For k=+1 (closed) r c −R 1 − 1 = t + c R2 2
2 2 R + (t + c2) = c1 (3.35) √ where the maximum value of the scale factor is Rmax = c1
20 • For k=-1 (open) r c R 1 + 1 = t + c R2 2
2 2 (t + c2) − R = c1 (3.36)
3.3.2 Euclidean regime
The respective differential equation for a Euclidean space is
c0 R˙ 2 − k = 1 (3.37) R2N where 1 N = 1 − 3z 2 and z is the barotropic index in the Euclidean regime. Thus, we see that by selecting an equation of state like this we get M = N = 1 (as stated in the previous section). That means that the solutions for k=+1 in the Euclidean spacetime will be the same as the solutions for k=-1 in the Lorentzian and vice versa. This statement holds for any equation of state of the form p = wµ. Integration now gives:
• For k=0 (flat) s p 0 R = 2 c1t + c2 (3.38)
• For k=+1 (open) 0 2 2 0 (t + c2) − R = c1 (3.39)
• For k=-1 (closed) 2 0 2 0 R + (t + c2) = c1 (3.40)
p 0 where Rmax = c1
3.3.3 Matching the solutions for k=+1 on both sides
We here assume that k=+1 on both sides so we are dealing with an open Euclidean spacetime followed by a closed Lorentzian spacetime. First we impose the condition R(0) = 0 on (13) so we get
0 02 c1 = c2
21 so the evolution of the scale factor is governed by
0 2 2 02 (t + c2) − R = c2
Next we must impose the junction conditions. We assume that the transition occurs at time t0.
Then, the first junction condition implies continuity of the scale factor at time t0 which is written as
lim R(t) = lim R(t) = R0 + − t→t0 t→t0 This can be imposed by taking the limits of (8) and (13) and then adding them to get
2 0 2 02 (t0 + c2) + (t0 + c2) = c1 + c2 (3.41)
The second junction condition is obtained by taking the jump of the quantities in the Friedmann equation (3) assuming that the density µ is continuous and thus its jump is zero. Taking into account the continuity of the scale factor, the second junction condition can be stated as
˙ 2 ˙ 2 R |L −R |E= −2 2 0 2 (t0 + c2) (t0 + c2) ⇒ 2 − 2 = −2 R0 R0 2 0 2 0 2 02 ⇒ (t0 + c2) − (t0 + c2) = −2[(t0 + c2) − c2 ] 2 0 2 02 ⇒ (t0 + c2) + (t0 + c2) = 2c2 02 ⇒ c1 = c2 where in the third line we used the expression for R0 given by (13) and in the last line we used the first junction condition (15). What we have obtained so far can be summarized in
0 02 c1 = c1 = c2 (3.42) which could have been also obtained by using the simplified expression of the second junction 0 condition which would immediately give us c1 = c1. However in this case the necessary algebra was trivial so we did the straightforward calculation to show that both ways give the same result indeed. For more complicated equations of R(t) it can be quite hard to obtain this result and the use of our simplified form is inevitable. Substituting c1 in (3.41) and solving for c2 we get q 02 2 c2 = −t0 ± c2 − R0 (3.43)
02 2 02 so we must require that c2 > R0 which is a very reasonable result considering that c2 = c1 and c1 is the maximum radius of the (closed) Lorentzian spacetime as can be seen from (3.35). So this condition ensures that when the transition happens, the radius of the Euclidean space will not be greater that the maximum radius of the Lorentzian manifold. Now in order to choose one of the solutions in (3.43) we must understand the meaning of the constant c2. Going back to p 02 2 equation (3.35) we see that at time t = −c2 = t0 ∓ c2 − R0 the scale factor takes its maximal
22 √ 0 value which is Rmax = c1 = −c2. And this time must follow t0 and not precede it. So the only possible solution for c2 is q 02 2 c2 = −t0 − c2 − R0 (3.44) Equations (3.42) and (3.44) are the equations we were seeking, reducing all the unknown 0 parameters in the arbitrary constant c2 which is the maximum radius of the final state. Using these equations we can have the final form of the scale factor as a function of time given by the equations
0 2 2 02 (t + c2) − R = c2 , t < t0 q 2 2 02 2 02 R + t − t0 − c2 − R0 = c2 , t > t0
0 where c2 can also be calculated in terms of R0 and t0 by simply taking the limit as t approaches t0 in the first one which results in 2 2 0 R0 − t0 c2 = 2t0 so all the unknowns are R0 and t0.
3.3.4 Matching the solutions for k=-1 on both sides
In this case we start with a closed Euclidean space (equation (3.40)) and the transition leads us to an open Lorentzian one (equation (3.36)). Again we begin by using the arbitrariness of the initial time, setting R(0) = 0 in (3.40) which gives
0 02 c1 = c2
so (3.40) can be rewritten as 2 0 2 02 R + (t + c2) = c2
0 we see here that the scale factor takes its maximum value for t = −c2 which is
0 0 Rmax = R(−c2) = c2
Again we will call the time of transition t0 and thus the first junction condition can be expressed again by taking the limits of (3.36) and (3.40) and adding them together to obtain
0 2 2 02 (t0 + c2) + (t0 + c2) = c1 + c2 (3.45)
The second junction condition is obtained very easily from our simplified expression:
0 c1 = c1
So just like the previous case we have so far attained
0 02 c1 = c1 = c2 (3.46)
23 Now we use the above in (3.45) and solve for c2 to get q 02 2 c2 = −t0 ± c2 + R0 (3.47)
In order to choose one of the solutions (3.47) we go back to equation (3.36) and see what happens for t = −c2 2 2 02 −R (−c2) = c1 ⇒ −R (−c2) = c2 which is impossible and so the time −c2 must not be contained in the domain of the Lorentzian function (3.36) or in other words it must be less than t0. This leaves us with the final solution for c2 q 02 2 c2 = −t0 + c2 + R0 (3.48)
So the final form of the scale factor equations is the following
2 0 2 02 R + (t + c2) = c2 , t < t0 q 02 2 2 2 02 (t − t0 + c2 + R0) − R = c2 , t > t0
0 In the above solution it might be physically interesting to set t = −c2 which corresponds to a transition when the scale factor in the Euclidean space reaches its maximum. In that case we 0 also have R0 = t0 = −c2 and the solution becomes
2 0 2 02 R + (t + c2) = c2 , t < t0 √ 0 2 2 02 (t + c2(1 + 2)) − R = c2 , t > t0
3.4 Application for inflationary fluid
In this section we will examine the evolution of a universe in which the berotropic fluid obeys the equation of state p = −µ
This kind of equation of state (the Lorentzian part of it) is the one used in Friedmann models to describe the inflation epoch. The main characteristic of such an equation of state is that it describes a fluid of constant density as can be easily seen from (3.29).
24 3.4.1 Lorentzian regime
Here M = −1 so our evolution equation gives:
˙ 2 R + k = c1R (3.49) so for the different values of k we obtain
• For k=+1 we get the necessary condition
2 c1R > 1 (3.50)
and under this condition we get the solution √ 1 p 2 √ ln(c1R + c1 c1R −) = t + c2 c1
√ √ e c1(c2+t) e− c1(t+c2) ⇒ R(t) = + (3.51) 2c1 2 • For k=-1 (3.49) gives −1 √ sinh ( c1R) √ = t + c2 c1 √ sinh[ c (t + c )] ⇒ R(t) = √1 2 (3.52) c1 • For k=0 (3.49) gives 1 √ ln R = t + c2 c1
√ ⇒ R(t) = e c1(t+c2) (3.53)
3.4.2 Euclidean regime
In the Euclidean regime N=-1 and (3.17) gives
• For k=+1 sinh[pc0 (t + c )] R(t) = 1 2 (3.54) p 0 c1
• For k=-1 √ √ c0 (c +t) − c0 (t+c ) e 1 2 e 1 2 R(t) = 0 + (3.55) 2c1 2 0 2 where the condition c1R (t) > 1 should hold.
• For k=0 √ R(t) = e c1(t+c2) (3.56)
25 3.4.3 Matching the solutions for k=+1
In this case we begin with a Euclidean regime whose evolution is governed by equation (3.54) and after the transition we end up in a Lorentzian regime whose scale factor is given by (3.51). 0 By imposing the condition R(0) = 0 (with no loss of generality) we get c2 = 0 so the equation in the Euclidean regime becomes: √ sinh( c t) R(t) = 1 (3.57) p 0 c1
0 By the second junction condition we get c1 = c1 so imposing the first junction condition gives us
√ √ p 0 c1(c2+t0) − c1(t0+c2) sinh( c1t0) e e √ = + = R0 (3.58) c1 2c1 2 which can be solved for c2 to give
p 2 2 ln c1R0 ± c1R0 − c1 c2 = −t0 + √ (3.59) c1 where c1 can also be expressed in terms of R0 and t0 as it satisfies √ sinh( c1t0) R0 = √ (3.60) c1 which can be solved for c1 with arbitrary accuracy but is not an algebraic equation. In order to determine which solution of (3.59) is the correct one, we first note the physical meaning of the constant c2. Calculating the time at which the Lorentzian universe is at its minimum, we find that this is p 2 2 ∗ ln c1 ln(c1R0 ± c1R0 − c1 t = √ + t0 − √ 2 c1 c1 and this time must not precede the time of transition t0, otherwise the solutions will not cross because the Euclidean solution will be always above the Lorentzian one as can be shown. If we demand this we get after some algebra q √ 2 1 > c1R0 ± c1R0 − 1 which holds only for the minus sign and this is why we accept only the solution of (3.59) with the minus sign. After this final step, we have everything we need. We have derived the solutions of the Friedmann equations and the junction conditions and we have found all the constants reducing the freedom of our solutions to the time of transition t0 and the value of the scale factor at this time R0 as we wished.
3.4.4 Matching the solutions for k=-1
In this case the solution in the Euclidean regime is given by (3.55) and the respective solution in 0 the Lorentzian regime by (3.52). In this case we cannot set R(0) = 0 but we can still set c2 = 0
26 (however in this case time does not start from zero and the equation for the scale factor has a 0 meaning for t → −∞). Imposing the second junction we again have c1 = c1 and so we proceed with the first condition. √ √ √ c1t0 − c1t0 sinh( c1(t0 + c2) e e √ = + = R0 (3.61) c1 2c1 2
From these we can solve for c2 as a function of c1,t0 and R0
√ −1 √ sinh( c1(t0 + c2) sinh c1R0 √ = R0 ⇒ c2 = √ − t0 (3.62) c1 c1 and c1 can be obtained as a function of R0 and t0 √ √ e c1t0 e− c1t0 + = R0 2c1 2 which can solved uniquely for c1 (using the same arguments as before to exclude one of the solutions). Again, we have everything we wanted, the solutions and the imposed conditions that leave us with only R0 and t0 as arbitrary constants.
27
Chapter 4
Conclusion
In the present thesis we have examined the properties of spacetimes with non-constant metric signature. We first examined the strict formalism that allows us to interchange between metrics with different signatures. This formalism gave us the form of the field equations as well as the necessary junction conditions that were to be imposed on the transition hypersurface. In order to achieve this formalism, we used the 1+3 foliation of spacetime and we demanded that the 3-d hypersurfaces (that represent ”space” in the Lorentzian spacetime) behave in a well defined way, in the sense that the geometric quantities defined on them are not ill-defined. Using the above arguments for the surface of transition, led us to the final form of our junction conditions that were used for matching the solutions. The freedom that we attained from the allowance of two coexisting metric signatures became evident in the form of the final solutions that we arrived at. More specifically, we examined universes with barotropic fluids and we focused on 2 special cases: the relativistic fluid and the inflationary fluid. In those universes we searched for interesting solutions with peculiar properties (compared to standard FRW models) by allowing the extrinsic curvature tensor to have a non-zero jump at the surface of transition. This statements, translated in the language of FRW universes, means that the derivative of the scale factor need not be continuous at the time of signature change. The above holds in universes where the curvature is non-zero. This trivial case of flat spacetime was partially ignored because it results in solutions with no interesting properties and has been presented in papers written in the past. Our solutions showed us new possibilities for the evolution of a universe. In the case of the relativistic fluid, we saw that the nature of the solutions in the Euclidean and Lorentzian regimes are completely different and can take us from potentially open universes (if the transition did not occur) to closed ones. In the case of the inflationary fluid, we showed that the extremum of the scale factor corresponds to a minimum (while in the case of a relativistic fluid it was a maximum). This gave us a universe that first went through a phase of contraction and the expanded for ever. Such properties are important in the sense that they could be used to solve theoretical problems in the future.
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Appendix A
The matter tensor
We are dealing with a barotropic fluid and the stress energy tensor will be assumed to take the perfect fluid form:
Tab = µuaub + phab ⇔ Tab = (µ + p)uaub + pgab (A.1) where p is the pressure of the fluid and µ its energy density (measured by an observer moving with the fluid). The following identities can be easily verified
a a c a d u Tab = −µub, h cT b = ph b,T = T d = −µ + 3p (A.2) and 1 µ = T uaub, p = habT (A.3) ab 3 ab a The eigenvalues of Tab are λ1 = −µ and λ2 = p and they correspond to the eigenvectors u and a a a ei , i = 1, 2, 3 respectively where ei are the basis vectors of the tangent space orthogonal to u . The conservation equations take the form:
µ˙ + (µ + p)Θ = 0 (A.4)
a ab (µ + p)u ˙ + h p,b = 0 (A.5)
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Appendix B
The field equations
We are here concerned with the field equations. The Einstein field equations are:
1 1 κT = R − Rg ⇔ R = κ T − T g (B.1) ab ab 2 ab ab ab 2 ab where 8πG R = Rd = κ(µ − 3p) and κ = d c4
Combining the above equations with Tab = (µ + p)uaub + pgab we acquire:
1 R = κ µu u + ph + µ − 3p g (B.2) ab a b ab 2 ab
Contracting (2) with uaub we get the (0,0) equation:
1 κ R uaub = κ µ + (−)(µ − 3p) = µ + 3p (B.3) ab 2 2
The (0, ν) equations are: a b Rabu h c = 0 (B.4) Proof: κ κ R ua = −κµu + µ − 3p u = − µ + 3p u ab b 2 b 2 b κ R uahb = − µ + 3p u δb + u ub = 0 ab c 2 b c c
Finally the (µ,ν) field equations are given by:
κ R ha hb = µ − p h (B.5) ab c d 2 cd
33 Proof: κ R ha = R (δa + u ua) = R + u R ua = R − u µ + 3p u ab c ab c c cb c ab cb c 2 b κ R ha hb = R hb − µ + 3p u u hb = R hb = ab c d cb d 2 c b d cb d κ R − µ + 3p u u = cd 2 c d 1 κ κ µu u + ph + µ − 3p h − u u − µ + 3p u u = c d cd 2 cd c d 2 c d µ 3 µ 3 κ κ κ µ − + p − − p u u + µ − p h = µ − p h 2 2 2 2 c d 2 cd 2 cd
34 Appendix C
The Friedmann equations
We here examine the kinematics and dynamics of FRW universes.By the assumption of isotropy many terms of the 1+3 decomposition disappear. More specifically we have that:
a u˙ = σab = ωab = 0 (C.1) so we are left with: 1 u = Θ(t)h (C.2) a;b 3 ab while it is hab is defined by:
hab = gab + uaub (C.3)
Finally the main equation we will work with is the Ricci identity:
d ua;bc − ua;cb = Rabcud (C.4)
From equations (2)and (3) it follows that
hab;c = (uaub);c = uaub;c + ubua;c = 1 1 1 Θh u + Θh u = Θ h u + h u 3 bc a 3 ac b 3 ac b bc a
Next we wish to calculate ua;bc to use it in the Ricci identity:
1 1 1 ua;bc = (ua;b);c = Θhab = habΘ;c + Θhab;c = 3 ;c 3 3 1 1 1 h Θ + Θ Θ h u + h u = 3 ab ,c 3 3 ac b bc a 1 h Θ + Θ2 h u + h u 3 ab ,c 9 ac b bc a
35 So we can now calculate the right hand side of the Ricci identity:
ua;bc − ua;cb = 1 1 h Θ + Θ2 h u + h u − h Θ − Θ2 h u + h u = 3 ab ,c 9 ac b bc a 3 ac ,b 9 ab c cb a 1 h Θ − h Θ + Θ2 h u − h u = 3 ab ,c ac ,b 9 ac b ab c − h Θ˙ u − h Θ˙ u + Θ2 h u − h u = 3 ab c ac b 9 ac b ab c 1 h u − h u Θ˙ + Θ2 3 ac b ab c 3
Plugging our result in the Ricci identity we get:
1 Rd u = h u − h u Θ˙ + Θ2 abc d 3 ac b ab c 3
Inverting the summation index and contracting with ub we get:
1 1 R ubud = − Θ˙ + Θ2 h dabc 3 3 ac
Contraction over a and c gives:
1 udR = Θ˙ + Θ2 u db 3 b
And finally contracting the above with ub we arrive at:
1 R ubud = − Θ˙ + Θ2 bd 3
From this last equation combined with the (0, 0) field equation
κ R uaub = µ + 3p ab 2 we get the Raychaudhuri equation:
1 R¨ κ Θ˙ + Θ2 ≡ 3 = µ + 3p (C.5) 3 R 2
36 where κ = 8πG/c4. Another useful equation can be deduced by the Gauss-Codazzi equation that relates the Rie- mann tensor of the three dimensional hypersurfaces with the projection of the four dimensional Riemann tensor of the space-time onto those hypersurfaces:
1 3R = R + Θ2 h h − h h (C.6) dabc ⊥ dabc 9 ab cd ac bd
Contracting with hbd and using the (µ, ν) equations (B.5) together with the Raychaudhuri equation (C.5) we have:
1 2 3R = R − Θ˙ + Θ2 h − Θ2h ac ⊥ ac 3 3 ac 9 ac 1 ⇔ 3R = R − Θ˙ + Θ2 h ac ⊥ ac 3 ac 2 1 ⇔ 3R = κµ − Θ2 h ac 3 3 ac
Contracting again with hac gives the Ricci scalar of these spaces:
1 3R = 2 κµ − Θ2 (C.7) 3
And if we now use the geometric equation 6k 3R = (C.8) a2 that relates the Ricci scalar of the 3-spaces with the scale factor, we finally get:
R˙ 2 κ k = µ − (C.9) R2 3 R2 which is sometimes individually refered to as Friedmann equation. Equations (C.5) and (C.9) together with the energy conservation (A.4) form the set of equations that we use to describe the evolution of the FRW universe. Those three equations are not independent but any two of them are.
37